I would suggest that Esther is burying much of her past construction of identity. In going to Joan's funeral and burying her, Esther is able to begin the process of reclaiming a new identity and new sense of self. This notion of identity is relatively free from the social notion of self that had been imposed upon her for so long. Having to deal with the weight of such expectation is another element that is buried along with Joan. In her wondering as to what she thought was being buried, it is clear that Esther sees a part of herself being buried. This portion of her identity could be the "old brag" of her "heart," the remnant of her past notion of self that is now being replaced with something new, something more authentic.
Joan's burial paves Esther's way "for being born twice—patched, retreaded and approved for the road." The deep breath that she takes prior to thinking about what is being buried is a part of this reflective process. Esther now understands how consciousness in the modern setting is a challenging element. It is something that requires adversity and a sense of resiliency. When she recognizes this at Joan's funeral, it is a moment in which her own identity is better understood. The social constructions of self that Esther had suffered under for so long is what she is burying. A new start is what emerges from this burial.